NEW YORK, March 13 (Reuters) - U.S. President Donald Trump's high-profile ouster of Secretary of State Rex Tillerson sent the U.S. dollar down slightly on Tuesday, with crude oil falling after an initial jump.

Trump gave Tillerson the boot on Tuesday after a series of public rifts over policy on North Korea, Russia and Iran, and replaced his chief diplomat with loyalist Central Intelligence Agency Director Mike Pompeo.

His firing - the biggest shakeup of Trump's Cabinet since he took office in January 2017 - was announced by the president on Twitter.

Evercore ISI strategist Stan Shipley in New York called it "another sign of a problem in the Trump administration."

The dollar index, which measures the greenback against a basket of major currencies, fell 0.28 percent, with the euro up 0.52 percent at $1.2396.

Crude oil initially jumped on the Tillerson news, with investors seeing signs that a deal on Iran's nuclear program could collapse, but the trend reversed later in the day.

MSCI's gauge of stocks across the globe shed 0.06 percent, while the pan-European FTSEurofirst 300 stock index lost 0.94 percent.

Analysts were largely unmoved by the departure of Tillerson, the latest in a cavalcade of exits from Trump's White House, and suggested its market impact could be short-lived.

David Kotok, chairman of money management firm Cumberland Advisors in Sarasota, Florida, said such turnover is "now regularly accepted for the markets."

"I dont think Tillerson out and a replacement is a shock to the market that lasts very much at all," Kotok said.

Separately, the U.S. Labor Department announced its Consumer Price Index rose 0.2 percent in February, in line with economists' expectations. The data suggests the Federal Reserve remains on track to raise interest rates at a gradual pace this year, which boosted some markets.

U.S. crude fell 1.21 percent to $60.62 per barrel and Brent was at $64.57, down 0.59 percent.

U.S. Treasuries retraced some early losses after choppy trading following the news on Tillerson. Benchmark 10-year notes last rose 4/32 in price to yield 2.8554 percent, from 2.87 percent late on Monday.

The consumer price data seemed to offset any negative impact of the Tillerson ouster on U.S. stocks, with Wall Street seesawing slightly in morning trading.

The Dow Jones Industrial Average rose 11.13 points, or 0.04 percent, to 25,189.74, the S&P 500 lost 2.28 points, or 0.08 percent, to 2,780.74 and the Nasdaq Composite dropped 32.05 points, or 0.42 percent, to 7,556.28.

European stocks fell across the board, with some analysts suggesting Tillerson's departure fueled concerns that extremists are gaining influence in Trump's administration.

Investec economist Philip Shaw pointed to fears that the move "will give the nationalists greater power within the White House over the globalists."